\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  -32.0 & +  8.00 x_{1} & +  5.00 x_{2} & +  6.00 x_{3} & -7.00 x_{4} & -5.00 x_{5} & +  3.00 x_{6} & -7.00 x_{7}\\
 x_{9}   &  29.0 & -9.00 x_{1} & +  2.00 x_{2} &    &   & +  2.00 x_{5} & +  6.00 x_{6} & -5.00 x_{7}\\
 x_{10}   &  16.0 & +  1.00 x_{1} & -2.00 x_{2} &   & +  2.00 x_{4} & -6.00 x_{5} & -9.00 x_{6} & -2.00 x_{7}\\
 x_{11}   &  -31.0 & -2.00 x_{1} & +  9.00 x_{2} & -7.00 x_{3} & -8.00 x_{4} & +  8.00 x_{5} & +  5.00 x_{6} & + 10.00 x_{7}\\
 x_{12}   &  -5.0 & +  3.00 x_{1} & +  6.00 x_{2} & -10.00 x_{3} & + 10.00 x_{4} & +  9.00 x_{5} & -6.00 x_{6} & +  9.00 x_{7}\\
 x_{13}   &  -24.0 & +  1.00 x_{1} & +  8.00 x_{2} & -4.00 x_{3} & + 10.00 x_{4} & +  3.00 x_{5} & +  4.00 x_{6} & +  2.00 x_{7}\\
\hline
z    &  0.0 & +  3.00 x_{1} & -5.00 x_{2} & -1.00 x_{3} & -4.00 x_{4} &   & +  2.00 x_{6} &   \\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -8.00 y_{8} & +  9.00 y_{9} & -1.00 y_{10} & +  2.00 y_{11} & -3.00 y_{12} & -1.00 y_{13}\\
 y_{2}   &  1.0 & -5.00 y_{8} & -2.00 y_{9} & +  2.00 y_{10} & -9.00 y_{11} & -6.00 y_{12} & -8.00 y_{13}\\
 y_{3}   &  1.0 & -6.00 y_{8} &    &   & +  7.00 y_{11} & + 10.00 y_{12} & +  4.00 y_{13}\\
 y_{4}   &  1.0 & +  7.00 y_{8} &   & -2.00 y_{10} & +  8.00 y_{11} & -10.00 y_{12} & -10.00 y_{13}\\
 y_{5}   &  1.0 & +  5.00 y_{8} & -2.00 y_{9} & +  6.00 y_{10} & -8.00 y_{11} & -9.00 y_{12} & -3.00 y_{13}\\
 y_{6}   &  1.0 & -3.00 y_{8} & -6.00 y_{9} & +  9.00 y_{10} & -5.00 y_{11} & +  6.00 y_{12} & -4.00 y_{13}\\
 y_{7}   &  1.0 & +  7.00 y_{8} & +  5.00 y_{9} & +  2.00 y_{10} & -10.00 y_{11} & -9.00 y_{12} & -2.00 y_{13}\\
\hline
z    &  -0 & + 32.00 y_{8} & -29.00 y_{9} & -16.00 y_{10} & + 31.00 y_{11} & +  5.00 y_{12} & + 24.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 5 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{8}   &  0.134146341463 & -0.11 y_{1} & +  0.94 y_{9} & -0.06 y_{10} & -0.02 y_{2} & -0.48 y_{12} & -0.30 y_{13}\\
 y_{11}   &  0.0365853658537 & +  0.06 y_{1} & -0.74 y_{9} & +  0.26 y_{10} & -0.10 y_{2} & -0.40 y_{12} & -0.72 y_{13}\\
 y_{3}   &  0.451219512195 & +  1.09 y_{1} & -10.84 y_{9} & +  2.16 y_{10} & -0.54 y_{2} & + 10.04 y_{12} & +  0.79 y_{13}\\
 y_{4}   &  2.23170731707 & -0.28 y_{1} & +  0.62 y_{9} & -0.38 y_{10} & -0.95 y_{2} & -16.55 y_{12} & -17.89 y_{13}\\
 y_{5}   &  1.37804878049 & -1.04 y_{1} & +  8.65 y_{9} & +  3.65 y_{10} & +  0.66 y_{2} & -8.16 y_{12} & +  1.23 y_{13}\\
 y_{6}   &  0.414634146341 & +  0.02 y_{1} & -5.10 y_{9} & +  7.90 y_{10} & +  0.56 y_{2} & +  9.44 y_{12} & +  0.51 y_{13}\\
 y_{7}   &  1.57317073171 & -1.38 y_{1} & + 19.01 y_{9} & -0.99 y_{10} & +  0.80 y_{2} & -8.30 y_{12} & +  3.06 y_{13}\\
\hline
z    &  5.42682926829 & -1.62 y_{1} & -22.01 y_{9} & -10.01 y_{10} & -3.80 y_{2} & -22.70 y_{12} & -8.06 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  1.62195121951 & +  0.11 x_{8} & -0.06 x_{11} & -1.09 x_{3} & +  0.28 x_{4} & +  1.04 x_{5} & -0.02 x_{6} & +  1.38 x_{7}\\
 x_{9}   &  22.012195122 & -0.94 x_{8} & +  0.74 x_{11} & + 10.84 x_{3} & -0.62 x_{4} & -8.65 x_{5} & +  5.10 x_{6} & -19.01 x_{7}\\
 x_{10}   &  10.012195122 & +  0.06 x_{8} & -0.26 x_{11} & -2.16 x_{3} & +  0.38 x_{4} & -3.65 x_{5} & -7.90 x_{6} & +  0.99 x_{7}\\
 x_{2}   &  3.80487804878 & +  0.02 x_{8} & +  0.10 x_{11} & +  0.54 x_{3} & +  0.95 x_{4} & -0.66 x_{5} & -0.56 x_{6} & -0.80 x_{7}\\
 x_{12}   &  22.6951219512 & +  0.48 x_{8} & +  0.40 x_{11} & -10.04 x_{3} & + 16.55 x_{4} & +  8.16 x_{5} & -9.44 x_{6} & +  8.30 x_{7}\\
 x_{13}   &  8.06097560976 & +  0.30 x_{8} & +  0.72 x_{11} & -0.79 x_{3} & + 17.89 x_{4} & -1.23 x_{5} & -0.51 x_{6} & -3.06 x_{7}\\
\hline
z    &  -5.42682926829 & -0.13 x_{8} & -0.04 x_{11} & -0.45 x_{3} & -2.23 x_{4} & -1.38 x_{5} & -0.41 x_{6} & -1.57 x_{7}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  1.62195121951 & +  0.11 x_{8} & -0.06 x_{11} & -1.09 x_{3} & +  0.28 x_{4} & +  1.04 x_{5} & -0.02 x_{6} & +  1.38 x_{7}\\
 x_{9}   &  22.012195122 & -0.94 x_{8} & +  0.74 x_{11} & + 10.84 x_{3} & -0.62 x_{4} & -8.65 x_{5} & +  5.10 x_{6} & -19.01 x_{7}\\
 x_{10}   &  10.012195122 & +  0.06 x_{8} & -0.26 x_{11} & -2.16 x_{3} & +  0.38 x_{4} & -3.65 x_{5} & -7.90 x_{6} & +  0.99 x_{7}\\
 x_{2}   &  3.80487804878 & +  0.02 x_{8} & +  0.10 x_{11} & +  0.54 x_{3} & +  0.95 x_{4} & -0.66 x_{5} & -0.56 x_{6} & -0.80 x_{7}\\
 x_{12}   &  22.6951219512 & +  0.48 x_{8} & +  0.40 x_{11} & -10.04 x_{3} & + 16.55 x_{4} & +  8.16 x_{5} & -9.44 x_{6} & +  8.30 x_{7}\\
 x_{13}   &  8.06097560976 & +  0.30 x_{8} & +  0.72 x_{11} & -0.79 x_{3} & + 17.89 x_{4} & -1.23 x_{5} & -0.51 x_{6} & -3.06 x_{7}\\
\hline
z    &  -14.1585365854 & +  0.21 x_{8} & -0.67 x_{11} & -6.94 x_{3} & -7.91 x_{4} & +  6.40 x_{5} & +  4.73 x_{6} & +  8.16 x_{7}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  1.62195121951 & +  0.11 x_{8} & -0.06 x_{11} & -1.09 x_{3} & +  0.28 x_{4} & +  1.04 x_{5} & -0.02 x_{6} & +  1.38 x_{7}\\
 x_{9}   &  22.012195122 & -0.94 x_{8} & +  0.74 x_{11} & + 10.84 x_{3} & -0.62 x_{4} & -8.65 x_{5} & +  5.10 x_{6} & -19.01 x_{7}\\
 x_{10}   &  10.012195122 & +  0.06 x_{8} & -0.26 x_{11} & -2.16 x_{3} & +  0.38 x_{4} & -3.65 x_{5} & -7.90 x_{6} & +  0.99 x_{7}\\
 x_{2}   &  3.80487804878 & +  0.02 x_{8} & +  0.10 x_{11} & +  0.54 x_{3} & +  0.95 x_{4} & -0.66 x_{5} & -0.56 x_{6} & -0.80 x_{7}\\
 x_{12}   &  22.6951219512 & +  0.48 x_{8} & +  0.40 x_{11} & -10.04 x_{3} & + 16.55 x_{4} & +  8.16 x_{5} & -9.44 x_{6} & +  8.30 x_{7}\\
 x_{13}   &  8.06097560976 & +  0.30 x_{8} & +  0.72 x_{11} & -0.79 x_{3} & + 17.89 x_{4} & -1.23 x_{5} & -0.51 x_{6} & -3.06 x_{7}\\
\hline
z    &  -14.1585365854 & +  0.21 x_{8} & -0.67 x_{11} & -6.94 x_{3} & -7.91 x_{4} & +  6.40 x_{5} & +  4.73 x_{6} & +  8.16 x_{7}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  4.26093088858 & -0.00 x_{8} & +  0.03 x_{11} & +  0.21 x_{3} & +  0.21 x_{4} & -0.12 x_{9} & +  0.59 x_{6} & -0.90 x_{7}\\
 x_{5}   &  2.54583921016 & -0.11 x_{8} & +  0.09 x_{11} & +  1.25 x_{3} & -0.07 x_{4} & -0.12 x_{9} & +  0.59 x_{6} & -2.20 x_{7}\\
 x_{10}   &  0.729196050776 & +  0.46 x_{8} & -0.57 x_{11} & -6.73 x_{3} & +  0.64 x_{4} & +  0.42 x_{9} & -10.05 x_{6} & +  9.01 x_{7}\\
 x_{2}   &  2.12834978843 & +  0.10 x_{8} & +  0.04 x_{11} & -0.29 x_{3} & +  1.00 x_{4} & +  0.08 x_{9} & -0.95 x_{6} & +  0.64 x_{7}\\
 x_{12}   &  43.4654442877 & -0.41 x_{8} & +  1.10 x_{11} & +  0.19 x_{3} & + 15.96 x_{4} & -0.94 x_{9} & -4.63 x_{6} & -9.63 x_{7}\\
 x_{13}   &  4.92524682652 & +  0.44 x_{8} & +  0.61 x_{11} & -2.34 x_{3} & + 17.98 x_{4} & +  0.14 x_{9} & -1.24 x_{6} & -0.35 x_{7}\\
\hline
z    &  2.14104372355 & -0.49 x_{8} & -0.12 x_{11} & +  1.09 x_{3} & -8.38 x_{4} & -0.74 x_{9} & +  8.51 x_{6} & -5.92 x_{7}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  4.28415758592 & +  0.01 x_{8} & +  0.01 x_{11} & -0.03 x_{10} & +  0.23 x_{4} & -0.11 x_{9} & +  0.27 x_{6} & -0.61 x_{7}\\
 x_{5}   &  2.68168482816 & -0.02 x_{8} & -0.02 x_{11} & -0.19 x_{10} & +  0.05 x_{4} & -0.04 x_{9} & -1.28 x_{6} & -0.52 x_{7}\\
 x_{3}   &  0.108340318525 & +  0.07 x_{8} & -0.08 x_{11} & -0.15 x_{10} & +  0.10 x_{4} & +  0.06 x_{9} & -1.49 x_{6} & +  1.34 x_{7}\\
 x_{2}   &  2.09702430847 & +  0.08 x_{8} & +  0.07 x_{11} & +  0.04 x_{10} & +  0.97 x_{4} & +  0.06 x_{9} & -0.52 x_{6} & +  0.26 x_{7}\\
 x_{12}   &  43.4863788768 & -0.40 x_{8} & +  1.09 x_{11} & -0.03 x_{10} & + 15.98 x_{4} & -0.93 x_{9} & -4.92 x_{6} & -9.38 x_{7}\\
 x_{13}   &  4.67204526404 & +  0.28 x_{8} & +  0.81 x_{11} & +  0.35 x_{10} & + 17.76 x_{4} & -0.00 x_{9} & +  2.25 x_{6} & -3.48 x_{7}\\
\hline
z    &  2.2590108969 & -0.41 x_{8} & -0.21 x_{11} & -0.16 x_{10} & -8.27 x_{4} & -0.67 x_{9} & +  6.88 x_{6} & -4.46 x_{7}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  4.30349375614 & +  0.02 x_{8} & -0.01 x_{11} & -0.06 x_{10} & +  0.24 x_{4} & -0.10 x_{9} & -0.18 x_{3} & -0.38 x_{7}\\
 x_{5}   &  2.58860670689 & -0.08 x_{8} & +  0.05 x_{11} & -0.06 x_{10} & -0.03 x_{4} & -0.09 x_{9} & +  0.86 x_{3} & -1.67 x_{7}\\
 x_{6}   &  0.0725410411113 & +  0.05 x_{8} & -0.06 x_{11} & -0.10 x_{10} & +  0.06 x_{4} & +  0.04 x_{9} & -0.67 x_{3} & +  0.90 x_{7}\\
 x_{2}   &  2.0594920724 & +  0.05 x_{8} & +  0.09 x_{11} & +  0.09 x_{10} & +  0.94 x_{4} & +  0.04 x_{9} & +  0.35 x_{3} & -0.21 x_{7}\\
 x_{12}   &  43.1296478182 & -0.62 x_{8} & +  1.37 x_{11} & +  0.46 x_{10} & + 15.67 x_{4} & -1.14 x_{9} & +  3.29 x_{3} & -13.78 x_{7}\\
 x_{13}   &  4.83541462046 & +  0.38 x_{8} & +  0.68 x_{11} & +  0.12 x_{10} & + 17.90 x_{4} & +  0.09 x_{9} & -1.51 x_{3} & -1.46 x_{7}\\
\hline
z    &  2.75810298863 & -0.10 x_{8} & -0.60 x_{11} & -0.85 x_{10} & -7.83 x_{4} & -0.38 x_{9} & -4.61 x_{3} & +  1.70 x_{7}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  3.72150835643 & +  0.04 x_{8} & -0.02 x_{11} & -0.05 x_{10} & +  0.25 x_{4} & -0.07 x_{9} & -0.37 x_{3} & +  0.22 x_{5}\\
 x_{7}   &  1.54942470815 & -0.05 x_{8} & +  0.03 x_{11} & -0.04 x_{10} & -0.02 x_{4} & -0.05 x_{9} & +  0.51 x_{3} & -0.60 x_{5}\\
 x_{6}   &  1.46065339716 & +  0.00 x_{8} & -0.03 x_{11} & -0.13 x_{10} & +  0.05 x_{4} & -0.01 x_{9} & -0.21 x_{3} & -0.54 x_{5}\\
 x_{2}   &  1.73838918283 & +  0.06 x_{8} & +  0.09 x_{11} & +  0.10 x_{10} & +  0.94 x_{4} & +  0.05 x_{9} & +  0.24 x_{3} & +  0.12 x_{5}\\
 x_{12}   &  21.7757621567 & +  0.05 x_{8} & +  0.93 x_{11} & +  0.94 x_{10} & + 15.95 x_{4} & -0.39 x_{9} & -3.79 x_{3} & +  8.25 x_{5}\\
 x_{13}   &  2.57008482405 & +  0.45 x_{8} & +  0.64 x_{11} & +  0.17 x_{10} & + 17.93 x_{4} & +  0.17 x_{9} & -2.26 x_{3} & +  0.88 x_{5}\\
\hline
z    &  5.39388594944 & -0.18 x_{8} & -0.55 x_{11} & -0.91 x_{10} & -7.87 x_{4} & -0.48 x_{9} & -3.73 x_{3} & -1.02 x_{5}\\
\end{array}\]
Final Dictionary
Solution:  5.39388594944
Num Pivots:  5
\end{document}
